Nonadditive Entropy Research Articles

Alleviating both H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_0$$\\end{document} and σ8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _8$$\\end{document} tensions in Tsallis cosmology

We present how Tsallis cosmology can alleviate both H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_0$$\\end{document} and σ8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _8$$\\end{document} tensions simultaneously. Such a modified cosmological scenario is obtained by the application of the gravity-thermodynamics conjecture, but using the non-additive Tsallis entropy, instead of the standard Bekenstein–Hawking one. Hence, one obtains modified Friedmann equations, with extra terms that depend on the new Tsallis exponent δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} that quantifies the departure from standard entropy. We show that for particular δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} choices we can obtain a phantom effective dark energy, which is known to be one of the sufficient mechanisms that can alleviate H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_0$$\\end{document} tension. Additionally, for the same parameter choice we obtain an increased friction term and an effective Newton’s constant smaller than the usual one, and thus the σ8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _8$$\\end{document} tension is also solved. These features act as a significant advantage of Tsallis modified cosmology.

Analogies and Relations between Non-Additive Entropy Formulas and Gintropy.

We explore formal similarities and mathematical transformation formulas between general trace-form entropies and the Gini index, originally used in quantifying income and wealth inequalities. We utilize the notion of gintropy introduced in our earlier works as a certain property of the Lorenz curve drawn in the map of the tail-integrated cumulative population and wealth fractions. In particular, we rediscover Tsallis' q-entropy formula related to the Pareto distribution. As a novel result, we express the traditional entropy in terms of gintropy and reconstruct further non-additive formulas. A dynamical model calculation of the evolution of Gini index is also presented.

Nonadditive Entropy Application to Detrended Force Sensor Data to Indicate Balance Disorder of Patients with Vestibular System Dysfunction.

The healthy function of the vestibular system (VS) is of vital importance for individuals to carry out their daily activities independently and safely. This study carries out Tsallis entropy (TE)-based analysis on insole force sensor data in order to extract features to differentiate between healthy and VS-diseased individuals. Using a specifically developed algorithm, we detrend the acquired data to examine the fluctuation around the trend curve in order to consider the individual's walking habit and thus increase the accuracy in diagnosis. It is observed that the TE value increases for diseased people as an indicator of the problem of maintaining balance. As one of the main contributions of this study, in contrast to studies in the literature that focus on gait dynamics requiring extensive walking time, we directly process the instantaneous pressure values, enabling a significant reduction in the data acquisition period. The extracted feature set is then inputted into fundamental classification algorithms, with support vector machine (SVM) demonstrating the highest performance, achieving an average accuracy of 95%. This study constitutes a significant step in a larger project aiming to identify the specific VS disease together with its stage. The performance achieved in this study provides a strong motivation to further explore this topic.

Black hole geometrothermodynamics and critical phenomena: A look from Tsallis entropy-based perspective

We analyze geometrothermodynamics of charged anti-de Sitter (AdS) black holes with a global monopole in the framework of Tsallis statistics. The latter arises from a non-additive generalization of Boltzmann–Gibbs entropy, which is still recovered as a particular sub-case. We examine the effects of Tsallis entropy on small-large black holes phase-transitions and critical exponents, comparing the results with the liquid–gas change of phase occurring in van der Waals fluids. We also discuss some physical constraints on Tsallis parameter and the impact on sparsity of BH radiation. Thermodynamic curvature behavior and the nature of microscopic interactions are then explored from a geometric perspective within Ruppeiner formalism on the S−P coordinate space. Similar considerations are extended to charged (2+1)-dimensional Banados–Teitelboim–Zanelli (BTZ) black holes by computing the entropy-pressure and entropy-volume corrected curvatures. Our analysis shows that Tsallis prescription has non-trivial implications on black hole geometrothermodynamics, emphasizing the role of non-additive entropy in the correspondence between black holes and van der Waals systems.

Improving the Estimation of the Occurrence Time of an Impending Major Earthquake Using the Entropy Change of Seismicity in Natural Time Analysis

This article is focused on a new procedure concerning a more accurate identification of the occurrence time of an impending major earthquake (EQ). Specifically, we first recapitulate that, as was recently shown [P. Varotsos et al., Communications in Nonlinear Science and Numerical Simulation 125 (2023) 107370], natural time analysis of seismicity supplemented with the non-additive Tsallis entropy Sq leads to a shortening of the time window of an impending major EQ. This has been shown for the Tohoku mega-EQ of magnitude M9 that occurred in Japan on 11 March 2011, which is the largest event ever recorded in Japan. Here, we also show that such a shortening of the time window of an impending mainshock can be achieved for major, but smaller EQs, of the order of M8 and M7. In particular, the following EQs are treated: the Chiapas M8.2 EQ, which is Mexico’s largest EQ for more than a century that took place on 7 September 2017 near the coast of Chiapas state in Mexico, the 19 September 2017 M7.1 EQ that occurred within the Mexican flat slab, and the M7.1 Ridgecrest EQ on 6 July 2019 in California.

A Note on the Connection between Non-Additive Entropy and h-Derivative.

In order to study as a whole a wide part of entropy measures, we introduce a two-parameter non-extensive entropic form with respect to the h-derivative, which generalizes the conventional Newton-Leibniz calculus. This new entropy, Sh,h', is proved to describe the non-extensive systems and recover several types of well-known non-extensive entropic expressions, such as the Tsallis entropy, the Abe entropy, the Shafee entropy, the Kaniadakis entropy and even the classical Boltzmann-Gibbs one. As a generalized entropy, its corresponding properties are also analyzed.

Characteristic Function of the Tsallis q-Gaussian and Its Applications in Measurement and Metrology

The Tsallis q-Gaussian distribution is a powerful generalization of the standard Gaussian distribution and is commonly used in various fields, including non-extensive statistical mechanics, financial markets and image processing. It belongs to the q-distribution family, which is characterized by a non-additive entropy. Due to their versatility and practicality, q-Gaussians are a natural choice for modeling input quantities in measurement models. This paper presents the characteristic function of a linear combination of independent q-Gaussian random variables and proposes a numerical method for its inversion. The proposed technique makes it possible to determine the exact probability distribution of the output quantity in linear measurement models, with the input quantities modeled as independent q-Gaussian random variables. It provides an alternative computational procedure to the Monte Carlo method for uncertainty analysis through the propagation of distributions.

Senses along Which the Entropy Sq Is Unique.

The Boltzmann-Gibbs-von Neumann-Shannon additive entropy SBG=-k∑ipilnpi as well as its continuous and quantum counterparts, constitute the grounding concept on which the BG statistical mechanics is constructed. This magnificent theory has produced, and will most probably keep producing in the future, successes in vast classes of classical and quantum systems. However, recent decades have seen a proliferation of natural, artificial and social complex systems which defy its bases and make it inapplicable. This paradigmatic theory has been generalized in 1988 into the nonextensive statistical mechanics-as currently referred to-grounded on the nonadditive entropy Sq=k1-∑ipiqq-1 as well as its corresponding continuous and quantum counterparts. In the literature, there exist nowadays over fifty mathematically well defined entropic functionals. Sq plays a special role among them. Indeed, it constitutes the pillar of a great variety of theoretical, experimental, observational and computational validations in the area of complexity-plectics, as Murray Gell-Mann used to call it. Then, a question emerges naturally, namely In what senses is entropy Sq unique? The present effort is dedicated to a-surely non exhaustive-mathematical answer to this basic question.

Medical Applications of Nonadditive Entropies.

The Boltzmann-Gibbs additive entropy SBG=-k∑ipilnpi and associated statistical mechanics were generalized in 1988 into nonadditive entropy Sq=k1-∑ipiqq-1 and nonextensive statistical mechanics, respectively. Since then, a plethora of medical applications have emerged. In the present review, we illustrate them by briefly presenting image and signal processings, tissue radiation responses, and modeling of disease kinetics, such as for the COVID-19 pandemic.

Tsallis q-Statistics in Seismology.

Non-extensive statistical mechanics (or q-statistics) is based on the so-called non-additive Tsallis entropy. Since its introduction by Tsallis, in 1988, as a generalization of the Boltzmann-Gibbs equilibrium statistical mechanics, it has steadily gained ground as a suitable theory for the description of the statistical properties of non-equilibrium complex systems. Therefore, it has been applied to numerous phenomena, including real seismicity. In particular, Tsallis entropy is expected to provide a guiding principle to reveal novel aspects of complex dynamical systems with catastrophes, such as seismic events. The exploration of the existing connections between Tsallis formalism and real seismicity has been the focus of extensive research activity in the last two decades. In particular, Tsallis q-statistics has provided a unified framework for the description of the collective properties of earthquakes and faults. Despite this progress, our present knowledge of the physical processes leading to the initiation of a rupture, and its subsequent growth through a fault system, remains quite limited. The aim of this paper was to provide an overview of the non-extensive interpretation of seismicity, along with the contributions of the Tsallis formalism to the statistical description of seismic events.

From the Boltzmann equation with non-local correlations to a standard non-linear Fokker-Planck equation

In this work, we study the formal connections between the non-linear Fokker-Planck Equation associated with the non-additive entropy and the Boltzmann Equation with the non-additive correlation functional. The collisional term following the q-algebra is adopted. In the derivation of the non-additive Fokker-Planck Equation, two constraints are imposed on the final result: i) that the entropic index q is a characteristic parameter of the non-additive systems with a value that does not change with time, and ii) that for q→1 a smooth transition for the standard Fokker-Planck Equation is obtained.

Cosmology and nonadditive entropy

Nonadditive entropies have been proposed as alternatives to understanding the thermodynamics of Black Holes. Moreover, it has been suggested that the difficulties of quantization of gravity are an indication that gravity is an emergent phenomenon, and therefore an entropic formulation is warranted. In this work, we explore cosmologies that come from nonadditive entropies. We focus our study on the cosmologies derived from entropies in superstatistics.

Tsallis cosmology and its applications in dark matter physics with focus on IceCube high-energy neutrino data

In this paper we employ a recent proposal of C. Tsallis and formulate the first law of thermodynamics for gravitating systems in terms of the extensive but non-additive entropy. We pay a particular attention to an integrating factor for the heat one-form and show that in contrast to conventional thermodynamics it factorizes into thermal and entropic part. Ensuing two laws of thermodynamics imply Tsallis cosmology, which is then subsequently used to address the observed discrepancy between current bound on the Dark Matter relic abundance and present IceCube data on high-energy neutrinos. To resolve this contradiction we keep the conventional minimal Yukawa-type interaction between standard model and Dark Matter particles but replace the usual Friedmann field equations with Tsallis-cosmology-based modified Friedmann equations. We show that when the Tsallis scaling exponent delta sim 1.57 (or equivalently, the holographic scaling exponent alpha sim 3.13) the aforementioned discrepancy disappears.

Non-Additive Entropy Composition Rules Connected with Finite Heat-Bath Effects.

Mathematical generalizations of the additive Boltzmann-Gibbs-Shannon entropy formula have been numerous since the 1960s. In this paper we seek an interpretation of the Rényi and Tsallis q-entropy formulas single parameter in terms of physical properties of a finite capacity heat-bath and fluctuations of temperature. Ideal gases of non-interacting particles are used as a demonstrating example.

Statistical mechanical approach of complex networks with weighted links

Systems that consist of many localized constituents interacting with each other can be represented by complex networks. Consistently, network science has become highly popular in vast fields focusing on natural, artificial and social systems. We numerically analyze the growth of d-dimensional geographic networks (characterized by the index α G ⩾ 0; d = 1, 2, 3, 4) whose links are weighted through a predefined random probability distribution, namely , w being the weight (w c ⩾ 0; τ > 0). In this model, each site has an evolving degree k i and a local energy (i = 1, 2, …, N) that depend on the weights of the links connected to it. Each newly arriving site links to one of the pre-existing ones through preferential attachment given by the probability , where d ij is the Euclidean distance between the sites. Short- and long-range interactions respectively correspond to α A/d > 1 and 0 ⩽ α A/d ⩽ 1; α A/d → ∞ corresponds to interactions between close neighbors, and α A/d → 0 corresponds to infinitely-ranged interactions. The site energy distribution p(ɛ) corresponds to the usual degree distribution p(k) as the particular instance (w c, τ) = (1, 0). We numerically verify that the corresponding connectivity distribution p(ɛ) converges, when α A/d → ∞, to the weight distribution P(w) for infinitely wide distributions (i.e. τ → ∞, ∀w c) as well as for w c → 0, ∀τ. Finally, we show that p(ɛ) is well approached by the q-exponential distribution , which optimizes the nonadditive entropy S q under simple constraints; q depends only on α A/d, thus exhibiting universality.

Transport Equation for Small Systems and Nonadditive Entropy

The nonadditive entropy introduced by Tsallis in 1988 has been used in different fields and generalizes the Boltzmann entropy, extending the possibilities of the application of the statistical methods developed in the context of Mechanics. Here, we investigate one of the last points of the theory that is still under discussion: the source term of the nonextensive transport equation. Based on a simple system, we show that the nonadditivity is a direct consequence of the phase space topology and derive the source term that leads to the nonextensive transport equation.

Complex network growth model: Possible isomorphism between nonextensive statistical mechanics and random geometry.

In the realm of Boltzmann-Gibbs statistical mechanics, there are three well known isomorphic connections with random geometry, namely, (i) the Kasteleyn-Fortuin theorem, which connects the λ → 1 limit of the λ-state Potts ferromagnet with bond percolation, (ii) the isomorphism, which connects the λ → 0 limit of the λ-state Potts ferromagnet with random resistor networks, and (iii) the de Gennes isomorphism, which connects the n → 0 limit of the n-vector ferromagnet with self-avoiding random walk in linear polymers. We provide here strong numerical evidence that a similar isomorphism appears to emerge connecting the energy q-exponential distribution ∝ e (with q = 4 / 3 and β ω = 10 / 3) optimizing, under simple constraints, the nonadditive entropy S with a specific geographic growth random model based on preferential attachment through exponentially distributed weighted links, ω being the characteristic weight.

Deformed Mathematical Objects Stemming from the q-Logarithm Function

Generalized numbers, arithmetic operators, and derivative operators, grouped in four classes based on symmetry features, are introduced. Their building element is the pair of q-logarithm/q-exponential inverse functions. Some of the objects were previously described in the literature, while others are newly defined. Commutativity, associativity, and distributivity, and also a pair of linear/nonlinear derivatives, are observed within each class. Two entropic functionals emerge from the formalism, and one of them is the nonadditive Tsallis entropy.

Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions.

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function , Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended to the complex plane z and conjectured that all nontrivial zeros are in the axis. The nonadditive entropy , where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function . It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as , which recover the number x for . The q-prime numbers are then defined as the q-natural numbers , where n is a prime number We show that, for any value of q, infinitely many q-prime numbers exist; for they diverge for increasing prime number, whereas they converge for ; the standard prime numbers are recovered for . For , we generalize the function as follows: (). We show that this function appears to diverge at , . Also, we alternatively define, for , and , which, for , generically satisfy , in variance with the case, where of course .

The inflationary era of the universe via Tsallis cosmology

In this paper, using the first law of thermodynamics (FLT), we derive a modification of the Friedmann equation from the nonadditive Tsallis entropy associated with the apparent horizon of the Friedmann–Robertson–Walker (FRW) Universe, where matter inside the apparent horizon is represented by a scalar field with a potential. We suggest that the inflationary era of the universe may phenomenologically consist of the two phases. We determined that the first phase is an inflate under the slow-roll condition known as exponential accelerated expansion, and the second one is an expansion phase containing the kinetic term of the scalar field, leading to a quintessential era. After expressing the observable inflation indices for both phases in terms of e-foldings number with choosing power-law scalar potential, we have compared the results with the latest Planck observation data. With some restrictions on the nonadditive Tsallis parameter and power-term of the potential, we observe that inflation can occur with the two phases. It is determined that while the nonadditive Tsallis parameter is positive in the slow-roll phase, it takes negative values in the kinetic energy dominating phase. Nonetheless, for the apparent horizon of the universe, an increasing entropy exists in the first phase, while a decrease in entropy is observed for the second phase, which indicates that the universe is ready for the creation of matter. In addition, a correlation between slow-roll and kinetic inflation is shown by determining the kinetic energy-potential relationship via the exponential term of the potential.